Circular Orbits

Newton's own picture: fire a cannonball fast enough horizontally and it falls, but the ground curves away beneath it just as fast. Keep increasing the speed and eventually the ball falls forever without hitting anything. That is orbit. There is no special anti-gravity force holding a satellite up; gravity is the only force acting, and it is pulling the satellite straight down the whole time. The satellite is in freefall. It just has so much sideways velocity that it falls around the planet instead of into it.

Because gravity is the only force, it has to do double duty: it has to be exactly the centripetal force required for circular motion at that radius. Setting GMm/r² equal to mv²/r and solving gives v = √(GM/r). Notice the mass of the satellite cancels, and notice the radius sits in the denominator inside a square root: larger r means smaller v. That is the opposite of the common gut feeling that higher orbits should feel faster or more energetic. A higher orbit is slower, but it still has more total mechanical energy, because the potential energy term -GMm/r climbs toward zero faster than the kinetic energy term falls. Total mechanical energy for a circular orbit works out to E = -GMm/(2r), exactly half the potential energy, always negative, meaning the satellite is gravitationally bound. This connects directly to angular momentum conservation and elliptical orbits: a circular orbit is the special case where speed and radius never change, while real orbits trade speed for radius according to the same gravity-supplies-centripetal-force logic applied at every point.